We present a mathematically closed, self-contained theory of boundary-driven diffusion with multiplicative noise on the half-line, which arises naturally as the hydrodynamic limit of the Dyson–Bures Stochastic Model (DBSM) in the Information Geometry of Emergent Spacetime (IGES). Using the Martin–Siggia–Rose–Janssen–De Dominicis (MSRJD) functional integral, we isolate a relevant boundary operator whose renormalization group (RG) flow is computed exactly at one loop with a heatkernel regularization, yielding the scheme-independent β-function β(g) = g− 4πD g 2 1 (D is the diffusion constant). The infrared-stable fixed point g ∗ = 4πD defines a universality class of hard-edge spectral overcrowding.We prove the following core theorem: (i) the stationary eigenvalue density obeys the hard-edge law ρ(λ) ∝ λ −1/2 ; (ii) the Stieltjes transform of ρ yields a fractional memory kernel K(t) ∝ t −1/2 ; (iii) the anomalous dimension of the boundary field is η = 1/2, fixed by the Callan–Symanzik equation. These results are derived without any free parameters or phenomenological fits. We then embed this rigorous core into the IGES framework: the DBSM provides the microscopic origin of the stochastic dynamics, the Mori–Zwanzig projection connects the spectral density to the cone-deviation memory kernel, and the effective fractional Langevin equation unifies four dynamical regimes—Vapour, Droplet, Boiling, and Ice-distinguished by the spectral gap ∆. The theory yields specific, falsifiable predictions for blackhole ringdown, including a t −3/2 power-law tail and a f −1/2 coloured noise floor, testable with LIGO/Virgo/KAGRA O5 via stacked residual analysis. The rigorous core is entirely independent of any spacetime interpretation; the gravitational phenomenology is an effective extension.
Mikheil Rusishvili (Fri,) studied this question.